Determine the amplitude, period, and displacement for the function y 4 The amplitude of the function is (Simplify your answer.) The period of the function is (Simplify your answer. Type an exact answer, using as needed.) The displacement of the function is. (Simplify your answer. Type an exact answer, using as needed.) Sketch the graph of the function. Choose the correct answer below. O A. Ау 1- OB. Ay 1 14 sin Q Then sketch the graph of the function. OC. 1+ O D. 1+
Determine the amplitude, period, and displacement for the function y 4 The amplitude of the function is (Simplify your answer.) The period of the function is (Simplify your answer. Type an exact answer, using as needed.) The displacement of the function is. (Simplify your answer. Type an exact answer, using as needed.) Sketch the graph of the function. Choose the correct answer below. O A. Ау 1- OB. Ay 1 14 sin Q Then sketch the graph of the function. OC. 1+ O D. 1+
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![To determine the amplitude, period, and displacement for the function \( y = \frac{1}{4} \sin \left( \frac{1}{5} x - \frac{\pi}{5} \right) \), follow these steps:
### Amplitude
The amplitude of the function is the coefficient in front of the sine function.
\[ \text{Amplitude} = \frac{1}{4} \]
(Simplify your answer.)
### Period
The period of the sine function is determined by the coefficient of \(x\) inside the sine function.
The formula to find the period \(T\) is:
\[ T = \frac{2\pi}{\left| \frac{1}{5} \right|} = 2\pi \times 5 = 10\pi \]
(Simplify your answer. Type an exact answer, using \(\pi\) as needed.)
### Displacement
The displacement (or phase shift) \(D\) can be found by setting the argument of the sine function equal to zero and solving for \(x\):
\[ \frac{1}{5} x - \frac{\pi}{5} = 0 \]
\[ \frac{1}{5} x = \frac{\pi}{5} \]
\[ x = \pi \]
So, the displacement is:
\[ \text{Displacement} = \pi \]
(Simplify your answer. Type an exact answer, using \(\pi\) as needed.)
### Graph
Sketch the graph of the function by choosing the correct graph from the options provided.
- **Option A:** Shows a sine wave starting at \(\pi\), amplitude of \(1\), period \(4\pi\).
- **Option B:** Shows a sine wave starting at \(\pi\), amplitude of \(1\), period \(4\pi\).
- **Option C:** Shows a sine wave starting at \(\pi\), amplitude of \(1\), period \(2\pi\).
- **Option D:** Shows a sine wave starting at \(\pi\), amplitude of \(1\), period \(2\pi\).
Among these, none depict the correct function \( y = \frac{1}{4} \sin \left( \frac{1}{5} x - \frac{\pi}{5} \right) \) accurately, considering the calculated period \(10\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7f7e91-311f-40b2-8ef2-058480eb16a0%2F149264c1-d163-4da5-a5ae-0d13e1edeac6%2Faa8wef9_processed.png&w=3840&q=75)
Transcribed Image Text:To determine the amplitude, period, and displacement for the function \( y = \frac{1}{4} \sin \left( \frac{1}{5} x - \frac{\pi}{5} \right) \), follow these steps:
### Amplitude
The amplitude of the function is the coefficient in front of the sine function.
\[ \text{Amplitude} = \frac{1}{4} \]
(Simplify your answer.)
### Period
The period of the sine function is determined by the coefficient of \(x\) inside the sine function.
The formula to find the period \(T\) is:
\[ T = \frac{2\pi}{\left| \frac{1}{5} \right|} = 2\pi \times 5 = 10\pi \]
(Simplify your answer. Type an exact answer, using \(\pi\) as needed.)
### Displacement
The displacement (or phase shift) \(D\) can be found by setting the argument of the sine function equal to zero and solving for \(x\):
\[ \frac{1}{5} x - \frac{\pi}{5} = 0 \]
\[ \frac{1}{5} x = \frac{\pi}{5} \]
\[ x = \pi \]
So, the displacement is:
\[ \text{Displacement} = \pi \]
(Simplify your answer. Type an exact answer, using \(\pi\) as needed.)
### Graph
Sketch the graph of the function by choosing the correct graph from the options provided.
- **Option A:** Shows a sine wave starting at \(\pi\), amplitude of \(1\), period \(4\pi\).
- **Option B:** Shows a sine wave starting at \(\pi\), amplitude of \(1\), period \(4\pi\).
- **Option C:** Shows a sine wave starting at \(\pi\), amplitude of \(1\), period \(2\pi\).
- **Option D:** Shows a sine wave starting at \(\pi\), amplitude of \(1\), period \(2\pi\).
Among these, none depict the correct function \( y = \frac{1}{4} \sin \left( \frac{1}{5} x - \frac{\pi}{5} \right) \) accurately, considering the calculated period \(10\
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